Singular limit laminations, Morse index, and positive scalar curvature

Positive Scalar Curvature

One of the striking initial applications of the Seiberg-Witten invariants was to give new obstructions to the existence of Riemannian metrics of positive scalar curvature on 4– manifolds. The vanishing of the Seiberg–Witten invariants of a manifold admitting such a metric may be viewed as a non-linear generalization of the classic conditions [12, 11] derived from the Dirac operator. If a manifo...

Positive Scalar Curvature and Surgery

06 Singular Minimal Hypersurfaces and Scalar Curvature

Finding obstructions to positive scalar curvature and getting structural insight is presently based on two competing approaches: one path which is most travelled works in the context of spin geometry and gives quite a direct link to topology (cf. [GL1-2] and [G]). The second, much less used but a priori more general method of attack analyzes minimal hypersurfaces within the manifold under consi...

N ov 2 00 8 METRICS OF POSITIVE SCALAR CURVATURE AND GENERALISED MORSE FUNCTIONS , PART 1

It is well known that isotopic metrics of positive scalar curvature are concordant. Whether or not the converse holds is an open question, at least in dimensions greater than four. We show that for a particular type of concordance, constructed using the surgery techniques of Gromov and Lawson, this converse holds in the case of closed simply connected manifolds of dimension at least five.

Positive Scalar Curvature with Symmetry

We show an equivariant bordism principle for constructing metrics of positive scalar curvature that are invariant under a given group action. Furthermore, we develop a new codimension2 surgery technique which removes singular strata from fixed point free S-manifolds while preserving equivariant positive scalar curvature. These results are applied to derive the following generalization of a resu...